How to create poly cylinders on all edges easily?
Youd could clone your mesh with Alt + D and add a wireframe modifier to it, that way the meshed are kept linked and in sync so modifications propagate.
Or as suggested by Samoth add the wireframe modifier without the option Replace Original so the underlaying mesh is kept visible

The Wireframe modifier does add a flat shaded grid with square section, so you may want to add a bevel modifier on top of it to make it smooth and cylindrical